2017/07/26 by Samir Adly, Loïc Bourdin, Adly, Samir +3 · 1 citation
Computer Science · Mathematics · #Contact Mechanics and Variational Inequalities #FOS: Mathematics #Mathematical Inequalities and Applications #Numerical methods in inverse problems #Optimization and Control (math.OC) #Optimization and Variational Analysis
paper · pdf · doi:10.48550/arxiv.1707.08509
openalex publication_date 2017/07/26 · openalex created_date 2022/09/15 · openalex updated_date 2026/07/28
The main result of the present theoretical paper is an original decomposition\nformula for the proximal operator of the sum of two proper, lower\nsemicontinuous and convex functions f and g. For this purpose, we introduce\na new operator, called f-proximal operator of g and denoted by\n\proxfg, that generalizes the classical notion. Then we prove the\ndecomposition formula \proxf+g = \proxf \∘\n\proxfg. After collecting several properties and characterizations\nof \proxfg, we prove that it coincides with the fixed points of a\ngeneralized version of the classical Douglas-Rachford operator. This\nrelationship is used for the construction of a weakly convergent algorithm that\ncomputes numerically this new operator \proxfg, and thus, from the\ndecomposition formula, allows to compute numerically \proxf+g. It\nturns out that this algorithm was already considered and implemented in\nprevious works, showing that \proxfg is already present (in a\nhidden form) and useful for numerical purposes in the existing literature.\nHowever, to the best of our knowledge, it has never been explicitly expressed\nin a closed formula and neither been deeply studied from a theoretical point of\nview. The present paper contributes to fill this gap in the literature. Finally\nwe give an illustration of the usefulness of the decomposition formula in the\ncontext of sensitivity analysis of linear variational inequalities of second\nkind in a Hilbert space.\n